Studies On Dynamics Of Several Epidemic Models With Nonlinear Incidence Rates

This paper studies the dynamical behaviors of several epidemic models with nonlinear incidence rates,and is divided into the four parts.In the first chapter,the research background and development of epidemic models are summarized,and the main results of the paper are introduced.In the second chapter,considering the difference of birth rate and death rate,an SEIRS(Susceptible,Exposed,Infective,Removed and Susceptible)model with strongly nonlinear incidence rate is established.The existence of the positive equilibria of the model is discussed,the local stability of the disease-free equilibrium of the model is analyzed,and the sufficient and necessary conditions for local stability of the positive equilibria are obtained by Routh-Hurwitz criterion.Furthermore,the sufficient conditions for the global stability of the disease-free equilibrium and the positive equilibria are given by Lyapunov function and Lyapunov-LaSalle theorem.Then,the existence of Hopf bifurcation and the stability of periodic solutions generated by Hopf bifurcation are discussed.Finally,our main results are illustrated by numerical simulation of some examples.In the third chapter,an SEIRS model with more general nonlinear incidence rate.Firstly,the basic reproduction number is obtained by the next generation matrix method.Then,the local stability and global stability of the disease-free equilibrium and the positive equilibria are discussed by Routh-Hurwitz criterion,Lyapunov function and Lyapunov-LaSalle theorem.Finally,numerical simulations of some examples are given to illustrate our main results.In the fourth chapter,the main results of the paper are summarized and the further researches are given.